Weak maximum principle for the p-Laplacian

For the equation $\Delta_p u = 0$ in $U$ ($U$ open and bounded), does a weak maximum principle hold? (The maximum and minimum occur on $\partial U$)? If yes, someone can indicate a book with the theorem?

Thanks in advance ( my english is horrible, sorry ... )

If you wanted to prove it from scratch, you would argue that a $p$-harmonic function is the unique minimizer of $p$-energy for its boundary values. Since the truncation by $\sup_{\partial U} u$ does not increase the energy and does not change the boundary values, it has to keep the function the same.
• @LeandroTavares Yes, by letting $v$ be constant function. – 40 votes Jul 23 '13 at 17:29
• Does it hold for $p=1$ ? – Hirak Mar 5 at 10:29